Logarithms - Basics

Logarithm 

Logarithm of a positive number x to the base a ( a is a positive number not equal to 1 ) is the power y to which the base a must be raised in order to produce the number x. 

log_ax =y because a^y=x     a > 0 and a ≠ 1

Logarithms properties:

a.) logarithm of a product: $\log_a(r \cdot s) = \log_a r + \log_a s$

b.) logarithm of a quotient: ${\log }_a\frac{r}{s}={\log }_{a}r-{\log }_{a}s$

c.) logarithm of a power: ${\log }_a(x^n)=n\cdot {\log }_{a}x$

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1. Rewrite to logarithm notation:

10² = 100      log₁₀ 100 = 2
4⁵ =1024       log₄1024 = 5
13⁰ = 1         log₁₃1 = 0
10^-3 = 0,001  log₁₀0,001 =-3
64^0,5 =8        log₆₄8 = 0,5
5^-2 = 0,04     log₅0,04=-2

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2. Solve and give a reason:

log₁₀1000  log₁₀1000 = 3 because 10³ = 1000
log₃81      log₃81 = 4 because 3⁴ = 81
log₂0,5      log₂0,5 =-1 because 2^-1 = 0,5
log₁₇1       log₁₇1 = 0  because 17⁰ =1
log₁₁11     log₁₁11 = 1 because 11¹ = 11
log₅0,2      log₅0,2 = -1 because5^-1 = 0,2
log₁₅0       log₁₅0 =  
log₅(-25)   log₅(-25) = n0  
log_0,40,4    log_0,40,4 = 1 because 0,4¹ =0,4
log₁49       log₁49 = 0 undefined 

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3. Determine x:

log₂x = 3      x = 2³ =8
log₁₀x = -4    x = 10^-4 = 0,0001
log₁₆x = 0,5   x =16^0,5= 4
log₂₀x =1      x = 20¹=20
log₂₅x = -0,5  x =25^-0,5 = 0,2
log_0.2487x = 0  x =0,2487⁰ =1

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4. Determine a:

log_a25 =2        a = 5
log_a81 = 4       a = 3
log_a100000 =5 a = 10
log_a512 = 3     a = 8
log_a0,01 = -2   a = 10
log_a5 = 0,5     a = 25
log_a36 = 2      a = 6
log_a64 = 1      a = 64

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5. Logarithmize following expressions (to the base a)

 

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6. Determine x:

 

 

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7. Enumerate the expression:

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8. Logarithmize the expression (to the base a):

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9. Enumerate the expression:

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10. Enumerate the expression:

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11. Use the decadic logarithm to solve the equation:

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12.Use the decadic logarithm to solve the equation:

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13.Use the decadic logarithm to solve the equation:

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14.During t = 50 hours the activity of radioactive sodium lowers to 1/10 of the initial value. Determine the half life of the nuclide using a natural logarithm.

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